Multiple Solutions

Pre-Calculus · Axiom Academy

LESSON Multiple Solutions to Trig Equations Why a single trig equation has infinitely many answers — and how the periodic curve crossing a line hands you every one. Solving means asking: where does the sine wave equal ? Draw the horizontal line and every place it cuts the curve is a solution. Because the wave repeats every , that line keeps cutting it — there is no last crossing. The equation we're reading off the graph Each crossing repeats every full turn Inside a single turn, why exactly two ? On the unit circle, is the x -coordinate. A vertical line x = k slices the circle at two heights — one above the axis, one below — so two angles share that cosine. Here we solve , and the negative value pushes both crossings into the left half (quadrants II and III). — the acute angle to the x -axis. Cosine is negative left half quadrants II and III. The vertical line meets the circle at exactly two points, mirror images across the x -axis. Their angles, and , are the only solutions in — a clean two-per-turn. To name every solution at once, take one solution and add the period over and over: for every integer n . The integer n counts how many full turns you've stepped — forward ( ) or backward ( ). Tangent is the exception. Its graph repeats every , not , so its solutions sit one apart . The animation solves : the base solution is , and stamping each time lands on , , — using here would skip half of them. 4. A Faster Wave, More Answers

This is the written version of the interactive lesson above. See the full Pre-Calculus course.